Geometry & Topology

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

Xiuxiong Chen, Song Sun, and Bing Wang

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove the existence of a Kähler–Einstein metric on a K–stable Fano manifold using the recent compactness result on Kähler–Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kähler–Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K–stability, assuming bounds on geometry.

Article information

Geom. Topol., Volume 22, Number 6 (2018), 3145-3173.

Received: 19 October 2015
Revised: 27 February 2018
Accepted: 27 March 2018
First available in Project Euclid: 29 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 14J45: Fano varieties

Kähler Ricci flow convergence uniqueness Fano manifold K–stability


Chen, Xiuxiong; Sun, Song; Wang, Bing. Kähler–Ricci flow, Kähler–Einstein metric, and K–stability. Geom. Topol. 22 (2018), no. 6, 3145--3173. doi:10.2140/gt.2018.22.3145.

Export citation


  • R J Berman, D Witt Nyström, Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons, preprint (2014)
  • E Calabi, X X Chen, The space of Kähler metrics, II, J. Differential Geom. 61 (2002) 173–193
  • X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, I: Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015) 183–197
  • X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, II: Limits with cone angle less than $2\pi$, J. Amer. Math. Soc. 28 (2015) 199–234
  • X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, III: Limits as cone angle approaches $2\pi$ and completion of the main proof, J. Amer. Math. Soc. 28 (2015) 235–278
  • X X Chen, W Y He, On the Calabi flow, Amer. J. Math. 130 (2008) 539–570
  • X Chen, W He, The Calabi flow on Kähler surfaces with bounded Sobolev constant, I, Math. Ann. 354 (2012) 227–261
  • X Chen, S Sun, Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics, Ann. of Math. 180 (2014) 407–454
  • X Chen, B Wang, Space of Ricci flows, I, Comm. Pure Appl. Math. 65 (2012) 1399–1457
  • X Chen, B Wang, The Kähler Ricci flow on Fano manifolds, I, J. Eur. Math. Soc. 14 (2012) 2001–2038
  • X Chen, B Wang, Space of Ricci flows, II, A: Moduli of singular Calabi–Yau spaces, Forum Math. Sigma 5 (2017) art. id. e32
  • X X Chen, B Wang, Space of Ricci flows, II, B: Weak compactness of the flow Adapted from arXiv:1405.6797 To appear in J. Diff. Geom.
  • T Darvas, The Mabuchi completion of the space of Kähler potentials, Amer. J. Math. 139 (2017) 1275–1313
  • T Darvas, W He, Geodesic rays and Kähler–Ricci trajectories on Fano manifolds, Trans. Amer. Math. Soc. 369 (2017) 5069–5085
  • V Datar, G Székelyhidi, Kähler–Einstein metrics along the smooth continuity method, Geom. Funct. Anal. 26 (2016) 975–1010
  • S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289–349
  • S K Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005) 453–472
  • S K Donaldson, Kähler metrics with cone singularities along a divisor, from “Essays in mathematics and its applications” (P M Pardalos, T M Rassias, editors), Springer (2012) 49–79
  • S K Donaldson, Stability, birational transformations and the Kahler–Einstein problem, from “Surveys in differential geometry” (H-D Cao, S-T Yau, editors), Surv. Differ. Geom. 17, International, Boston (2012) 203–228
  • S Donaldson, S Sun, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math. 213 (2014) 63–106
  • S Donaldson, S Sun, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Differential Geom. 107 (2017) 327–371
  • R Feng, H Huang, The global existence and convergence of the Calabi flow on $\mathbb{C}^n/\mathbb{Z}^n+i\mathbb{Z}^n$, J. Funct. Anal. 263 (2012) 1129–1146
  • A Futaki, T Mabuchi, Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann. 301 (1995) 199–210
  • V Georgoulas, J W Robbin, D A Salamon, The moment-weight inequality and the Hilbert–Mumford criterion, preprint (2013)
  • U Görtz, T Wedhorn, Algebraic geometry, I: Schemes with examples and exercises, Vieweg + Teubner (2010)
  • W He, On the convergence of the Calabi flow, Proc. Amer. Math. Soc. 143 (2015) 1273–1281
  • H Li, B Wang, K Zheng, Regularity scales and convergence of the Calabi flow, To appear in J. Geom. Anal. (online publication July 2017)
  • S T Paul, Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics, Ann. of Math. 175 (2012) 255–296
  • S T Paul, A numerical criterion for K–energy maps of algebraic manifolds, preprint (2012)
  • S T Paul, Stable pairs and coercive estimates for the Mabuchi functional, preprint (2013)
  • D H Phong, N Sesum, J Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007) 613–632
  • N Sesum, G Tian, Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman), J. Inst. Math. Jussieu 7 (2008) 575–587
  • J Streets, Long time existence of minimizing movement solutions of Calabi flow, Adv. Math. 259 (2014) 688–729
  • S Sun, Y Wang, On the Kähler–Ricci flow near a Kähler–Einstein metric, J. Reine Angew. Math. 699 (2015) 143–158
  • G Székelyhidi, Extremal metrics and $K$–stability, Bull. Lond. Math. Soc. 39 (2007) 76–84
  • G Székelyhidi, The Kähler–Ricci flow and $K$–polystability, Amer. J. Math. 132 (2010) 1077–1090
  • G Székelyhidi, Filtrations and test-configurations, Math. Ann. 362 (2015) 451–484
  • G Székelyhidi, The partial $C^0$–estimate along the continuity method, J. Amer. Math. Soc. 29 (2016) 537–560
  • G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1–37
  • G Tian, Z Zhang, Regularity of Kähler–Ricci flows on Fano manifolds, Acta Math. 216 (2016) 127–176
  • G Tian, X Zhu, A new holomorphic invariant and uniqueness of Kähler–Ricci solitons, Comment. Math. Helv. 77 (2002) 297–325
  • V Tosatti, Kähler–Ricci flow on stable Fano manifolds, J. Reine Angew. Math. 640 (2010) 67–84
  • V Tosatti, B Weinkove, The Calabi flow with small initial energy, Math. Res. Lett. 14 (2007) 1033–1039
  • D Witt Nyström, Test configurations and Okounkov bodies, Compos. Math. 148 (2012) 1736–1756
  • S-T Yau, Open problems in geometry, from “Differential geometry: partial differential equations on manifolds” (R Greene, editor), Proc. Sympos. Pure Math. 54, Amer. Math. Soc., Providence, RI (1993) 1–28